Hypsometric equation explained

The hypsometric equation, also known as the thickness equation, relates an atmospheric pressure ratio to the equivalent thickness of an atmospheric layer considering the layer mean of virtual temperature, gravity, and occasionally wind. It is derived from the hydrostatic equation and the ideal gas law.

Formulation

The hypsometric equation is expressed as:^[1] $$h\; =\; z\_2\; -\; z\_1\; =\; \backslash frac\; \backslash ,\; \backslash ln\; \backslash left(\backslash frac\backslash right),$$where:

h = thickness of the layer [m],z = geometric height [m],R = specific gas constant for dry air,\overline{T _{v}} = mean virtual temperature in Kelvin [K],g = gravitational acceleration [m/s²],p = pressure [Pa].

In meteorology,

p _{1} and p _{2} are isobaric surfaces. In radiosonde observation, the hypsometric equation can be used to compute the height of a pressure level given the height of a reference pressure level and the mean virtual temperature in between. Then, the newly computed height can be used as a new reference level to compute the height of the next level given the mean virtual temperature in between, and so on.

Derivation

The hydrostatic equation:

p = \rho ⋅ g ⋅ z,

where

\rho is the density [kg/m³], is used to generate the equation for hydrostatic equilibrium, written in differential form:dp = - \rho ⋅ g ⋅ dz.

This is combined with the ideal gas law:

p = \rho ⋅ R ⋅ T _{v}

to eliminate

\rho :(d p)/(p) = (- g)/(R ⋅ T _{v}) d z.

This is integrated from

z _{1} to z _{2} :\int ^{p(z _{2)}} _{p(z _{1)}} (d p)/(p) = \int ^{z _{2}} _{z _{1}} (- g)/(R ⋅ T _{v}) d z.

R and g are constant with z, so they can be brought outside the integral.If temperature varies linearly with z (e.g., given a small change in z),it can also be brought outside the integral when replaced with

\overline{T _{v}} , the average virtual temperature between z _{1} and z _{2} .\int ^{p(z _{2)}} _{p(z _{1)}} (d p)/(p) = (- g)/(R ⋅ \overline{T _{v}) }\int_^ \, \mathrmz.

Integration gives

ln \left( (p(z _{2)})/(p(z _{1)}) \right) = (- g)/(R ⋅ \overline{T _{v}) } (z_2 - z_1),

simplifying to

ln \left( (p _{1})/(p _{2}) \right) = (g)/(R ⋅ \overline{T _{v}) } (z_2 - z_1).

Rearranging:

z _{2} - z _{1} = (R ⋅ \overline{T _{v})/() } \ln \left(\frac \right),

or, eliminating the natural log:

(p _{1})/(p _{2}) = e ^{(g)/(R ⋅ \overline{T _{v})} ⋅ (z _{2} - z _{1)}.}

Correction

The Eötvös effect can be taken into account as a correction to the hypsometric equation. Physically, using a frame of reference that rotates with Earth, an air mass moving eastward effectively weighs less, which corresponds to an increase in thickness between pressure levels, and vice versa. The corrected hypsometric equation follows:^[2] $$h\; =\; z\_2\; -\; z\_1\; =\; \backslash frac\; \backslash cdot\; \backslash ln\; \backslash left(\backslash frac\backslash right),$$where the correction due to the Eötvös effect, A, can be expressed as follows:$$A\; =\; -\backslash frac\; \backslash left(2\; \backslash Omega\; \backslash overline\; \backslash cos\; \backslash phi\; +\; \backslash frac\backslash right),$$where

\Omega = Earth rotation rate,\phi = latitude,r = distance from Earth center to the air mass,\overline{u} = mean velocity in longitudinal direction (east-west), and\overline{v} = mean velocity in latitudinal direction (north-south).

This correction is considerable in tropical large-scale atmospheric motion.

See also

• Barometric formula
• Vertical pressure variation

Notes and References

1. Web site: Hypsometric equation - AMS Glossary . American Meteorological Society . 12 March 2013.
2. Ong . H. . Roundy . P.E. . Nontraditional hypsometric equation . Q. J. R. Meteorol. Soc. . 2019 . 146 . 727 . 700–706 . 10.1002/qj.3703. free . 2011.09576 .